1, 3, 4, 8, 15, 27,?

The pattern is + 1, + 3, + 6,....., i.e. + 1, + (1, + 2), + (1 + 2 + 3),.....

So, missing term = 15 + (1 + 2 + 3 + 4) = 25.

The pattern is x 2 + 1, x 2 - 1, x 2 + 1, x 2 - 1,.....

So, missing term = 101 x 2 + 1 = 203.

The pattern is x 2, x 3/2, x 2, x 3/2, x 2,.....

So, missing term = 18 x 3/2 = 27.

The series consists of squares and cubes of consecutive natural numbers i.e. 1², 1³, 2², 2³, 3², 3³, 4², .....

So, missing term = 4³ = 64.

The pattern is - 45, - 35, - 25, .....

So, missing term = 20 - 15 = 5.

The given sequence is a combination of two series :

I. 3, 7, 13, 21, 31, ? and II. 4, 7, 13, 22, 34

The pattern in I is + 4, + 6, + 8, + 10,.....

The pattern in II is + 3, + 6, + 9, + 12,.....

So, missing term = 31 + 12 = 43.

The pattern is - 4, - 9, - 16,.....i.e. - 2², - 3², - 4²,.....

So, missing pattern = 169 - 5² = 169 - 25 = 144.

Clearly, the given series consists of prime numbers starting from 2. So, the missing

term is the prime number after 11, which is 13.

The pattern is + 6, + 9, + 12, + 15, .....

So, missing term = 21 + 12 = 33.

Clearly, 5 x 2 = 10, 10 x 2 = 20, 20 x 2 = 40,.....

So, the series is a G.P. in which a = 5 and r = 2.

Let 1280 be the rath term of the series.

Then, 5x2^n-1 = 1280 ⟺ 2^n-1 = 256 = 2⁸ ⟺ n - 1 = 8 ⟺ n = 9.